Cramér-Rao Bounds and Non-Linear Least Squares for a Seven Parameter Dual Channel Sinewave Model

The Cramer-Rao bound (CRB) is a lower bound on the error variance of any estimator. For a Gaussian scenario, the CRB is derived for a seven parameter dual channel sine wave model; a model with relevance in applications like impedance measurements and the estimation of particle size and velocity by laser anemometry. Three different parameterizations are considered: the common quadrature/in-phase and amplitude-phase models, as well as a relative amplitude-phase model. The CRB indicates the achievable error variance of an unbiased estimator as a function of the signal-to-noise ratio, number of samples, and noise power. An non-optimal non-linear least squares fit of the signal model to the collected data is employed. The problem at hand is separable and can be solved by a one dimensional search followed by a linear least-squares fit of the remaining parameters. The performance of the method is investigated by aid of a simulation study, and the outcome is compared with the corresponding CRB. For high signal-to-noise ratios, the performance of the proposed method is close to optimal with an error variance close to the predictions made by the CRB.